Asymmetric Bidirectional Predictive Dynamics in Google Search and COVID-19 Time Series: A Multi-Horizon Forecast-Error Analysis
Abstract: Online search activity is widely used as a digital trace of information seeking during epidemics, but its predictive value is often reduced to correlation, a selected lag, or a single forecast-error score. We examine eight pre-specified Google Trends-COVID-19 pairs in both directions using matched LightGBM forecasts over rolling 400-day windows and horizons of 1-14 days. Log-RMSE gain is combined with forecast-error distribution entropy (FEDE), which captures horizon-dependent changes in the signed error distribution. The resulting directional predictive signature comprises response class, RMSE effect, FEDE endpoint and early-slope responses, persistence across rolling regimes, and horizon profile. For screening candidate directions, we also introduce a relative persistence-weighted directional predictive response strength (DPRS) on a within-study 0-100 scale; 100 denotes the strongest observed response in the comparison set, not an absolute coupling maximum. The pairs show concordant, RMSE-only, FEDE-only, and mixed responses. "loss of smell" -> reported cases is accuracy-dominant, with mean log-RMSE gain 0.104 (about 9.9% equivalent RMSE reduction) and DPRS 100, whereas the reverse direction is FEDE-dominant (D_FEDE=0.048; DPRS=42). "covid test" -> positive rate is FEDE-dominant (D_FEDE=0.039; positive in 72% of windows; DPRS=98) despite negative RMSE gain, while "lockdown" -> stringency is the clearest concordant case (DPRS=78). World-level symptom/case pairs are less persistent. Thus, forecast accuracy and error-distribution dynamics are complementary, predictive effects can be direction- and metric-dependent, and forecast horizon is part of the relationship. Aggregated regime-level patterns should not be extrapolated to individual time points or interpreted as evidence of causality.